Theorem subgmulg 13793

Description: A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)

Hypotheses

Ref	Expression
subgmulgcl.t	|- .x. = (.g ` G )
subgmulg.h	|- H = ( G ↾s S )
subgmulg.t	|- .xb = (.g ` H )

Assertion

Ref	Expression
subgmulg	|- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )

Proof of Theorem subgmulg

Step	Hyp	Ref	Expression
1		subgmulg.h	. . . . . 6 |- H = ( G ↾s S )
2		eqid 2231	. . . . . 6 |- ( 0g ` G ) = ( 0g ` G )
3	1, 2	subg0 13785	. . . . 5 |- ( S e. (SubGrp ` G ) -> ( 0g ` G ) = ( 0g ` H ) )
4	3	3ad2ant1 1044	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( 0g ` G ) = ( 0g ` H ) )
5	4	ifeq1d 3623	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
6	1	a1i 9	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> H = ( G ↾s S ) )
7		eqid 2231	. . . . . . . . . . . 12 |- ( +g ` G ) = ( +g ` G )
8	7	a1i 9	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` G ) )
9		id 19	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> S e. (SubGrp ` G ) )
10		subgrcl 13784	. . . . . . . . . . 11 |- ( S e. (SubGrp ` G ) -> G e. Grp )
11	6, 8, 9, 10	ressplusgd 13230	. . . . . . . . . 10 |- ( S e. (SubGrp ` G ) -> ( +g ` G ) = ( +g ` H ) )
12	11	3ad2ant1 1044	. . . . . . . . 9 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( +g ` G ) = ( +g ` H ) )
13	12	seqeq2d 10717	. . . . . . . 8 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
14	13	adantr 276	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
15	14	fveq1d 5641	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) )
16	15	ifeq1d 3623	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) )
17		simprl 531	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> -. N = 0 )
18		simprr 533	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> -. 0 < N )
19		simp2 1024	. . . . . . . . . . . 12 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> N e. ZZ )
20		ztri3or0 9521	. . . . . . . . . . . 12 |- ( N e. ZZ -> ( N < 0 \/ N = 0 \/ 0 < N ) )
21	19, 20	syl 14	. . . . . . . . . . 11 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N < 0 \/ N = 0 \/ 0 < N ) )
22	21	adantr 276	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( N < 0 \/ N = 0 \/ 0 < N ) )
23	17, 18, 22	ecase23d 1386	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> N < 0 )
24		simpl1 1026	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> S e. (SubGrp ` G ) )
25	19	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> N e. ZZ )
26	25	znegcld 9604	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> -u N e. ZZ )
27	19	zred 9602	. . . . . . . . . . . . . . 15 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> N e. RR )
28	27	lt0neg1d 8695	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N < 0 <-> 0 < -u N ) )
29	28	biimpa 296	. . . . . . . . . . . . 13 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> 0 < -u N )
30		elnnz 9489	. . . . . . . . . . . . 13 |- ( -u N e. NN <-> ( -u N e. ZZ /\ 0 < -u N ) )
31	26, 29, 30	sylanbrc 417	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> -u N e. NN )
32		eqid 2231	. . . . . . . . . . . . . . . 16 |- ( Base ` G ) = ( Base ` G )
33	32	subgss 13779	. . . . . . . . . . . . . . 15 |- ( S e. (SubGrp ` G ) -> S C_ ( Base ` G ) )
34	33	3ad2ant1 1044	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> S C_ ( Base ` G ) )
35		simp3 1025	. . . . . . . . . . . . . 14 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. S )
36	34, 35	sseldd 3228	. . . . . . . . . . . . 13 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. ( Base ` G ) )
37	36	adantr 276	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> X e. ( Base ` G ) )
38		subgmulgcl.t	. . . . . . . . . . . . 13 |- .x. = (.g ` G )
39		eqid 2231	. . . . . . . . . . . . 13 |- seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` G ) , ( NN X. { X } ) )
40	32, 7, 38, 39	mulgnn 13731	. . . . . . . . . . . 12 |- ( ( -u N e. NN /\ X e. ( Base ` G ) ) -> ( -u N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) )
41	31, 37, 40	syl2anc 411	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( -u N .x. X ) = ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) )
42	35	adantr 276	. . . . . . . . . . . 12 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> X e. S )
43	38	subgmulgcl 13792	. . . . . . . . . . . 12 |- ( ( S e. (SubGrp ` G ) /\ -u N e. ZZ /\ X e. S ) -> ( -u N .x. X ) e. S )
44	24, 26, 42, 43	syl3anc 1273	. . . . . . . . . . 11 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( -u N .x. X ) e. S )
45	41, 44	eqeltrrd 2309	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) e. S )
46		eqid 2231	. . . . . . . . . . 11 |- ( invg ` G ) = ( invg ` G )
47		eqid 2231	. . . . . . . . . . 11 |- ( invg ` H ) = ( invg ` H )
48	1, 46, 47	subginv 13786	. . . . . . . . . 10 |- ( ( S e. (SubGrp ` G ) /\ ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) e. S ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
49	24, 45, 48	syl2anc 411	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ N < 0 ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
50	23, 49	syldan 282	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) )
51	13	adantr 276	. . . . . . . . . 10 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) )
52	51	fveq1d 5641	. . . . . . . . 9 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) = ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) )
53	52	fveq2d 5643	. . . . . . . 8 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` H ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
54	50, 53	eqtrd 2264	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ ( -. N = 0 /\ -. 0 < N ) ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
55	54	anassrs 400	. . . . . 6 |- ( ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) /\ -. 0 < N ) -> ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) = ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) )
56		0z 9490	. . . . . . 7 |- 0 e. ZZ
57	19	adantr 276	. . . . . . 7 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> N e. ZZ )
58		zdclt 9557	. . . . . . 7 |- ( ( 0 e. ZZ /\ N e. ZZ ) -> DECID 0 < N )
59	56, 57, 58	sylancr 414	. . . . . 6 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> DECID 0 < N )
60	55, 59	ifeq2dadc 3637	. . . . 5 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) )
61	16, 60	eqtrd 2264	. . . 4 |- ( ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) /\ -. N = 0 ) -> if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) = if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) )
62		0zd 9491	. . . . 5 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> 0 e. ZZ )
63		zdceq 9555	. . . . 5 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> DECID N = 0 )
64	19, 62, 63	syl2anc 411	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> DECID N = 0 )
65	61, 64	ifeq2dadc 3637	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
66	5, 65	eqtrd 2264	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
67	32, 7, 2, 46, 38, 39	mulgval 13727	. . 3 |- ( ( N e. ZZ /\ X e. ( Base ` G ) ) -> ( N .x. X ) = if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
68	19, 36, 67	syl2anc 411	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = if ( N = 0 , ( 0g ` G ) , if ( 0 < N , ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` G ) ` ( seq 1 ( ( +g ` G ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
69	1	subgbas 13783	. . . . 5 |- ( S e. (SubGrp ` G ) -> S = ( Base ` H ) )
70	69	3ad2ant1 1044	. . . 4 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> S = ( Base ` H ) )
71	35, 70	eleqtrd 2310	. . 3 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> X e. ( Base ` H ) )
72		eqid 2231	. . . 4 |- ( Base ` H ) = ( Base ` H )
73		eqid 2231	. . . 4 |- ( +g ` H ) = ( +g ` H )
74		eqid 2231	. . . 4 |- ( 0g ` H ) = ( 0g ` H )
75		subgmulg.t	. . . 4 |- .xb = (.g ` H )
76		eqid 2231	. . . 4 |- seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) = seq 1 ( ( +g ` H ) , ( NN X. { X } ) )
77	72, 73, 74, 47, 75, 76	mulgval 13727	. . 3 |- ( ( N e. ZZ /\ X e. ( Base ` H ) ) -> ( N .xb X ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
78	19, 71, 77	syl2anc 411	. 2 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .xb X ) = if ( N = 0 , ( 0g ` H ) , if ( 0 < N , ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` N ) , ( ( invg ` H ) ` ( seq 1 ( ( +g ` H ) , ( NN X. { X } ) ) ` -u N ) ) ) ) )
79	66, 68, 78	3eqtr4d 2274	1 |- ( ( S e. (SubGrp ` G ) /\ N e. ZZ /\ X e. S ) -> ( N .x. X ) = ( N .xb X ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 841   \/ w3o 1003   /\ w3a 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ifcif 3605   {csn 3669   class class class wbr 4088   X. cxp 4723   ` cfv 5326  (class class class)co 6018   0cc0 8032   1c1 8033   < clt 8214   -ucneg 8351   NNcn 9143   ZZcz 9479   seqcseq 10710   Basecbs 13100   ↾_s cress 13101   +g cplusg 13178   0gc0g 13357   Grpcgrp 13601   invgcminusg 13602  ._gcmg 13724  SubGrpcsubg 13772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-inn 9144  df-2 9202  df-n0 9403  df-z 9480  df-uz 9756  df-seqfrec 10711  df-ndx 13103  df-slot 13104  df-base 13106  df-sets 13107  df-iress 13108  df-plusg 13191  df-0g 13359  df-mgm 13457  df-sgrp 13503  df-mnd 13518  df-grp 13604  df-minusg 13605  df-mulg 13725  df-subg 13775

This theorem is referenced by:  zringmulg  14631